| L(s) = 1 | + 3-s − 10·5-s + 9·9-s − 3·11-s + 65·13-s − 10·15-s − 67·17-s + 168·19-s − 60·23-s + 75·25-s + 44·27-s − 151·29-s + 22·31-s − 3·33-s + 82·37-s + 65·39-s + 374·41-s − 142·43-s − 90·45-s + 723·47-s − 67·51-s + 288·53-s + 30·55-s + 168·57-s + 696·59-s − 474·61-s − 650·65-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.894·5-s + 1/3·9-s − 0.0822·11-s + 1.38·13-s − 0.172·15-s − 0.955·17-s + 2.02·19-s − 0.543·23-s + 3/5·25-s + 0.313·27-s − 0.966·29-s + 0.127·31-s − 0.0158·33-s + 0.364·37-s + 0.266·39-s + 1.42·41-s − 0.503·43-s − 0.298·45-s + 2.24·47-s − 0.183·51-s + 0.746·53-s + 0.0735·55-s + 0.390·57-s + 1.53·59-s − 0.994·61-s − 1.24·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.672009471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.672009471\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T - 8 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 2602 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 p T + 5388 T^{2} - 5 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 67 T + 7898 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 168 T + 19778 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 60 T + 24238 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 151 T + 36488 T^{2} + 151 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 22 T + 57462 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 82 T + 102738 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 374 T + 170570 T^{2} - 374 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 142 T + 143886 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 723 T + 324322 T^{2} - 723 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 288 T + 197974 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 696 T + 483058 T^{2} - 696 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 474 T + 468050 T^{2} + 474 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 302 T - 128898 T^{2} - 302 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 492 T + 695662 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 118 T + 775290 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 253 T + 593658 T^{2} + 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1674 T + 1539118 T^{2} + 1674 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 576 T + 1094482 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1383 T + 1853762 T^{2} + 1383 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.893998201534942054890731078843, −8.649317488977837990463970308294, −8.297413127381509654478317231877, −7.78683829915771117976244138724, −7.40218243279221805425255948959, −7.25948951997779583114755191444, −6.75166957522420200289234437026, −6.28132300197317978354243610398, −5.64196459701137273161550441292, −5.61079344102868747033391210249, −4.99572892651150186913766558283, −4.27526007303975128807849125632, −4.00599734273580776378257049356, −3.87921729474511949078609999174, −3.04125859101178628117829264431, −2.86559825314840933816546654450, −2.11072352653353356962206835460, −1.47548121901470705369831843386, −0.885412856196535910092059938048, −0.50012987723641486257158925613,
0.50012987723641486257158925613, 0.885412856196535910092059938048, 1.47548121901470705369831843386, 2.11072352653353356962206835460, 2.86559825314840933816546654450, 3.04125859101178628117829264431, 3.87921729474511949078609999174, 4.00599734273580776378257049356, 4.27526007303975128807849125632, 4.99572892651150186913766558283, 5.61079344102868747033391210249, 5.64196459701137273161550441292, 6.28132300197317978354243610398, 6.75166957522420200289234437026, 7.25948951997779583114755191444, 7.40218243279221805425255948959, 7.78683829915771117976244138724, 8.297413127381509654478317231877, 8.649317488977837990463970308294, 8.893998201534942054890731078843
